Ordered Compactifications with Countable Remainders

Countable compactifications of topological spaces have been studied in [1], [5], [7], and [9]. In [7], Magill showed that a locally compact, T2 topological space X has a countable T2 compactification if and only if it has n-point T2 compactifications for every integer n ≥ 1. We generalize this theorem to T2-ordered compactifications of ordered topological spaces. Before starting our generalization of Magill’s theorem, we recall two unpleasant facts about ordered compactifications. For the class of ordered topological spaces which allow T2-ordered compactifications (i.e., the T3.5ordered spaces), local compactness does not guarantee the existence of finitepoint T2-ordered compactifications (think of the reals with the usual order and discrete topology); furthermore the existence of an n-point T2-ordered compactification for some n > 1 does not guarantee the existence of a onepoint T2-ordered compactification (think of the reals with the usual order and topology). Here is our main theorem: If a T3.5-ordered space X allows a finite-point T2-ordered compactification, then X allows a countable T2ordered compactification if and only if there is a positive integer m such that X allows an n-point T2-ordered compactification for every n ≥ m. In case the order on X is equality, the result is equivalent to Magill’s theorem. An ordered topological space, or simply an ordered space is a triple (X, τ, θ) where τ is a topology on the set X and θ is the graph of a partial order on X. An ordered space (X, τ, θ) is T2-ordered if θ is closed in the product space X ×X, and is T3.5-ordered (completely regular ordered in [10])